# American Institute of Mathematical Sciences

June  2013, 6(2): 269-290. doi: 10.3934/krm.2013.6.269

## Nonlinear stability of a Vlasov equation for magnetic plasmas

 1 UPMC-Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu F75252 Paris cedex 05, France, France, France 2 INRIA Rocquencourt, BANG project-team, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France

Received  April 2012 Revised  November 2012 Published  February 2013

The mathematical description of laboratory fusion plasmas produced in Tokamaks is still challenging. Complete models for electrons and ions, as Vlasov-Maxwell systems, are computationally too expensive because they take into account all details and scales of magneto-hydrodynamics. In particular, for most of the relevant studies, the mass electron is negligible and the velocity of material waves is much smaller than the speed of light. Therefore it is useful to understand simplified models. Here we propose and study one of those which keeps both the complexity of the Vlasov equation for ions and the Hall effect in Maxwell's equation. Based on energy dissipation, a fundamental physical property, we show that the model is nonlinear stable and consequently prove existence.
Citation: Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic & Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269
##### References:
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Mischler, About the splitting algorithm for Boltzmann and B.G.K. equations, Mathematical Models and Methods in Applied Sciences, 6 (1996), 1079-1101. doi: 10.1142/S0218202596000444.  Google Scholar [12] R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.  Google Scholar [13] R. J. DiPerna, P.-L. Lions and Y. Meyer, Lp regularity of velocity averages, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 8 (1991), 271-287.  Google Scholar [14] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar [15] J. Freidberg, "Plasma Physics and Fusion Energy," Cambridge, 2007. Google Scholar [16] J.-F. Gerbeau, C. Le Bris and T. Lelièvre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals," Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.  Google Scholar [17] P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution, Kinetic and Related Models, 2 (2009), 707-725. doi: 10.3934/krm.2009.2.707.  Google Scholar [18] F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125. doi: 10.1016/0022-1236(88)90051-1.  Google Scholar [19] D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425. doi: 10.1080/03605302.2011.555804.  Google Scholar [20] P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson System, Inventiones Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.  Google Scholar [21] H. Lütjens and J.-F. Luciani, The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas, Journal of Computational Physics, 227 (2008), 6944-6966. doi: 10.1016/j.jcp.2008.04.003.  Google Scholar [22] H. Lütjens and J.-F. Luciani, XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks, Journal of Computational Physics, 229 (2010), 8130-8143. doi: 10.1016/j.jcp.2010.07.013.  Google Scholar [23] C. Mouhot and C. Villani, On Landau damping, Acta Mathematica, 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9.  Google Scholar [24] B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244. doi: 10.1090/S0273-0979-04-01004-3.  Google Scholar [25] B. Perthame and P.-E. Souganidis, A limiting case for velocity averaging, Annales Scientifiques de l' École Normale Supérieure, 31 (1998), 591-598. doi: 10.1016/S0012-9593(98)80108-0.  Google Scholar [26] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.  Google Scholar [27] R. Temam, Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations, 2 (1977), 563-585.  Google Scholar [28] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," American Mathematical Society, 2000. Google Scholar [29] J. Simon, Compact sets in the space $L^p(0, T ; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

show all references

##### References:
 [1] M. Acheritogaray, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinetic and Related Models, 4 (2011), 901-918. doi: 10.3934/krm.2011.4.901.  Google Scholar [2] J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics. With Application to Tokamaks", Wiley/Gauthier-Villars Series in Modern Applied Mathematics, John Wiley & sons, Ltd., Chichester, Gauthier-Villars, Montrouge, 1989.  Google Scholar [3] S.-I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Consultants Bureau, 1, New York, (1965), 205-311. Google Scholar [4] H. Brezis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité dans les plasmas, C. R. Acad. Sciences Paris Série I Math., 321 (1995), 953-959.  Google Scholar [5] F. Bouchut, F. Golse and M. Pulvirenti, "Kinetic Equations and Asymptotic Theory," Series in Appl. Math. (Paris), 4, Gauthiers-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000.  Google Scholar [6] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Math. Sciences, 106, Springer-Verlag, New York, 1994.  Google Scholar [7] F. Chen, "Introduction To Plasma Physics and Controlled Fusion," Springer, New-York, 1984. Google Scholar [8] P. Crispel, P. Degond and M.-H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, Journal of Computational Physics, 205 (2005), 408-438. doi: 10.1016/j.jcp.2004.11.011.  Google Scholar [9] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Sciences and Technology," Springer, 1990. Google Scholar [10] B. Després and R. Sart, Reduced resistive MHD in Tokamaks with general density, ESAIM Math. Model. Numer. Anal., 46 (2012), 1081-1106. doi: 10.1051/m2an/2011078.  Google Scholar [11] L. Desvillettes and S. Mischler, About the splitting algorithm for Boltzmann and B.G.K. equations, Mathematical Models and Methods in Applied Sciences, 6 (1996), 1079-1101. doi: 10.1142/S0218202596000444.  Google Scholar [12] R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.  Google Scholar [13] R. J. DiPerna, P.-L. Lions and Y. Meyer, Lp regularity of velocity averages, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 8 (1991), 271-287.  Google Scholar [14] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar [15] J. Freidberg, "Plasma Physics and Fusion Energy," Cambridge, 2007. Google Scholar [16] J.-F. Gerbeau, C. Le Bris and T. Lelièvre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals," Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.  Google Scholar [17] P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution, Kinetic and Related Models, 2 (2009), 707-725. doi: 10.3934/krm.2009.2.707.  Google Scholar [18] F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125. doi: 10.1016/0022-1236(88)90051-1.  Google Scholar [19] D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425. doi: 10.1080/03605302.2011.555804.  Google Scholar [20] P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson System, Inventiones Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.  Google Scholar [21] H. Lütjens and J.-F. Luciani, The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas, Journal of Computational Physics, 227 (2008), 6944-6966. doi: 10.1016/j.jcp.2008.04.003.  Google Scholar [22] H. Lütjens and J.-F. Luciani, XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks, Journal of Computational Physics, 229 (2010), 8130-8143. doi: 10.1016/j.jcp.2010.07.013.  Google Scholar [23] C. Mouhot and C. Villani, On Landau damping, Acta Mathematica, 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9.  Google Scholar [24] B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244. doi: 10.1090/S0273-0979-04-01004-3.  Google Scholar [25] B. Perthame and P.-E. Souganidis, A limiting case for velocity averaging, Annales Scientifiques de l' École Normale Supérieure, 31 (1998), 591-598. doi: 10.1016/S0012-9593(98)80108-0.  Google Scholar [26] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.  Google Scholar [27] R. Temam, Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations, 2 (1977), 563-585.  Google Scholar [28] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," American Mathematical Society, 2000. Google Scholar [29] J. Simon, Compact sets in the space $L^p(0, T ; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar
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